Bradford started well and Mark Yeates put the ball over early on after James Hanson had set him up following a cross by Kyel Reid.
Midfielder Mark Thompson should have put the Bantams ahead in the 14th minute when he wastefully shot over following Yeates' pass.
Crawley striker Jamie Proctor was twice denied by visiting goalkeeper Jon McLaughlin before Thompson failed to make the most of another good opportunity for Bradford.
Thompson found himself in a good position after Nathan Doyle headed on, but chipped the ball over from 10 yards.
Crawley winger Nicky Adams threatened to break the deadlock eight minutes before half-time after being found by Josh Simpson, but his goal-bound shot was deflected round the post by debutant defender Matthew Bates.
Former Peterborough striker Sinclair put Crawley ahead eight minutes after the break with a neat right-footed finish under McLaughlin after racing on to Andy Drury's pass.
Proctor later shot wide following an error by McLaughlin and Drury also went close when his deep cross struck the top of the crossbar.
Crawley goalkeeper Paul Jones saved brilliantly from Reid before McLaughlin dived to parry a goal-bound shot from Sinclair.